The Stacks project

109.51 A torsor which is not an fppf torsor

In Groupoids, Remark 39.11.5 we raise the question whether any $G$-torsor is a $G$-torsor for the fppf topology. In this section we show that this is not always the case.

Let $k$ be a field. All schemes and stacks are over $k$ in what follows. Let $G \to \mathop{\mathrm{Spec}}(k)$ be the group scheme

\[ G = (\mu _{2, k})^\infty = \mu _{2, k} \times _ k \mu _{2, k} \times _ k \mu _{2, k} \times _ k \ldots = \mathop{\mathrm{lim}}\nolimits _ n (\mu _{2, k})^ n \]

where $\mu _{2, k}$ is the group scheme of second roots of unity over $\mathop{\mathrm{Spec}}(k)$, see Groupoids, Example 39.5.2. As an inverse limit of affine schemes we see that $G$ is an affine group scheme. In fact it is the spectrum of the ring $k[t_1, t_2, t_3, \ldots ]/(t_ i^2 - 1)$. The multiplication map $m : G \times _ k G \to G$ is on the algebra level given by $t_ i \mapsto t_ i \otimes t_ i$.

We claim that any $G$-torsor over $k$ is of the form

\[ P = \mathop{\mathrm{Spec}}(k[x_1, x_2, x_3, \ldots ]/(x_ i^2 - a_ i)) \]

for certain $a_ i \in k^*$ and with $G$-action $G \times _ k P \to P$ given by $x_ i \to t_ i \otimes x_ i$ on the algebra level. We omit the proof. Actually for the example we only need that $P$ is a $G$-torsor which is clear since over $k' = k(\sqrt{a_1}, \sqrt{a_2}, \ldots )$ the scheme $P$ becomes isomorphic to $G$ in a $G$-equivariant manner. Note that $P$ is trivial if and only if $k' = k$ since if $P$ has a $k$-rational point then all of the $a_ i$ are squares.

We claim that $P$ is an fppf torsor if and only if the field extension $k' = k(\sqrt{a_1}, \sqrt{a_2}, \ldots )/k$ is finite. If $k'$ is finite over $k$, then $\{ \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)\} $ is an fppf covering which trivializes $P$ and we see that $P$ is indeed an fppf torsor. Conversely, suppose that $P$ is an fppf $G$-torsor. This means that there exists an fppf covering $\{ S_ i \to \mathop{\mathrm{Spec}}(k)\} $ such that each $P_{S_ i}$ is trivial. Pick an $i$ such that $S_ i$ is not empty. Let $s \in S_ i$ be a closed point. By Varieties, Lemma 33.14.1 the field extension $\kappa (s)/k$ is finite, and by construction $P_{\kappa (s)}$ has a $\kappa (s)$-rational point. Thus we see that $k \subset k' \subset \kappa (s)$ and $k'$ is finite over $k$.

To get an explicit example take $k = \mathbf{Q}$ and $a_ i = i$ for example (or $a_ i$ is the $i$th prime if you like).

Lemma 109.51.1. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. The stack $G\textit{-Principal}$ classifying principal homogeneous $G$-spaces (see Examples of Stacks, Subsection 94.14.5) and the stack $G\textit{-Torsors}$ classifying fppf $G$-torsors (see Examples of Stacks, Subsection 94.14.8) are not equivalent in general.

Proof. The discussion above shows that the functor $G\textit{-Torsors} \to G\textit{-Principal}$ isn't essentially surjective in general. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04AF. Beware of the difference between the letter 'O' and the digit '0'.